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Recent Developments in Discrete Convex Analysis

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Book cover Research Trends in Combinatorial Optimization

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This paper describes recent developments in discrete convex analysis. Particular emphasis is laid on natural introduction of the classes of L-convex and M-convex functions in discrete and continuous variables. Expansion of the application areas is demonstrated by recent connections to submodular function maximization, finite metric space, eigenvalues of Hermitian matrices, discrete fixed point theorem, and matching games.

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References

  • Ageev, A., Sviridenko, M.: Pipage rounding: A new method of constructing algorithms with proven performance guarantee. J. Comb. Optim. 8, 307–328 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Altman, E., Gaujal, B., Hordijk, A.: Multimodularity, convexity, and optimization properties. Math. Oper. Res. 25, 324–347 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Altman, E., Gaujal, B., Hordijk, A.: Discrete-Event Control of Stochastic Networks: Multimodularity and Regularity. Lecture Notes in Mathematics, vol. 1829. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  • Bandelt, H.-J., Dress, A.W.M.: A canonical decomposition theory for metrics on a finite set. Adv. Math. 92, 47–105 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Bouchet, A., Cunningham, W.H.: Delta-matroids, jump systems, and bisubmodular polyhedra. SIAM J. Discrete Math. 8, 17–32 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Buneman, P.: The recovery of trees from measures of dissimilarity. In: Hodson, R.F., Kendall, D.G., Tautu, P. (eds.) Mathematics in the Archaeological and Historical Sciences, pp. 387–395. Edinburgh University Press, Edinburgh (1971)

    Google Scholar 

  • Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint (extended abstract). In: Fischetti, M., Williamson, D.P. (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 4513, pp. 182–196. Springer, Berlin (2007)

    Chapter  Google Scholar 

  • Chen, X., Deng, X.: A simplicial approach for discrete fixed point theorems. In: Chen, D.Z., Lee, D.T. (eds.) Computing and Combinatorics. Lecture Notes in Computer Science, vol. 4112, pp. 3–12. Springer, Berlin (2006)

    Chapter  Google Scholar 

  • Conforti, M., Cornuéjols, G.: Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado–Edmonds theorem. Discrete Appl. Math. 7, 251–274 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. Wiley, New York (1998)

    MATH  Google Scholar 

  • Danilov, V.I., Koshevoy, G.A.: Discrete convexity and Hermitian matrices. Proc. Steklov Inst. Math. 241, 58–78 (2003)

    Google Scholar 

  • Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Springer, Berlin (1997)

    MATH  Google Scholar 

  • Dress, A.W.M., Wenzel, W.: Valuated matroid: A new look at the greedy algorithm. Appl. Math. Lett. 3, 33–35 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Dress, A.W.M., Wenzel, W.: Valuated matroids. Adv. Math. 93, 214–250 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Dress, A.W.M., Moulton, V., Terhalle, W.: T-theory: an overview. Eur. J. Comb. 17, 161–175 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Edmonds, J.: Submodular functions, matroids and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970). Also in: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization—Eureka, You Shrink! Lecture Notes in Computer Science, vol. 2570, pp. 11–26. Springer, Berlin (2003)

    Google Scholar 

  • Eriksson, K., Karlander, J.: Stable matching in a common generalization of the marriage and assignment models. Discrete Math. 217, 135–156 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Favati, P., Tardella, F.: Convexity in nonlinear integer programming. Ric. Oper. 53, 3–44 (1990)

    Google Scholar 

  • Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions II. Math. Program. Study 8, 73–87 (1978)

    MathSciNet  Google Scholar 

  • Fleiner, T.: A matroid generalization of the stable matching polytope. In: Gerards, B., Aardal, K. (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 2081, pp. 105–114. Springer, Berlin (2001)

    Google Scholar 

  • Fleiner, T.: A fixed point approach to stable matchings and some applications. Math. Oper. Res. 28, 103–126 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Frank, A.: A weighted matroid intersection algorithm. J. Algorithms 2, 328–336 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • Frank, A.: An algorithm for submodular functions on graphs. Ann. Discrete Math. 16, 97–120 (1982)

    MATH  Google Scholar 

  • Fujishige, S.: Theory of submodular programs: A Fenchel-type min-max theorem and subgradients of submodular functions. Math. Program. 29, 142–155 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Annals of Discrete Mathematics, vol. 58. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  • Fujishige, S., Murota, K.: Notes on L-/M-convex functions and the separation theorems. Math. Program. 88, 129–146 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Fujishige, S., Tamura, A.: A two-sided discrete-concave market with possibly bounded side payments: An approach by discrete convex analysis. Math. Oper. Res. 32, 136–155 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)

    MATH  Google Scholar 

  • Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull., New Ser., Am. Math. Soc. 37, 209–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Gale, D., Shapley, L.S.: College admissions and stability of marriage. Am. Math. Mon. 69, 9–15 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  • Hajek, B.: Extremal splittings of point processes. Math. Oper. Res. 10, 543–556 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Hirai, H.: A geometric study of the split decomposition. Discrete Comput. Geom. 36, 331–361 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Hirai, H., Murota, K.: M-convex functions and tree metrics. Jpn. J. Ind. Appl. Math. 21, 391–403 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Iimura, T.: A discrete fixed point theorem and its applications. J. Math. Econ. 39, 725–742 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Iimura, T., Murota, K., Tamura, A.: Discrete fixed point theorem reconsidered. J. Math. Econ. 41, 1030–1036 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Iwata, S.: Submodular function minimization. Math. Program. Ser. B 112, 45–64 (2007)

    Article  MathSciNet  Google Scholar 

  • Iwata, S., Shigeno, M.: Conjugate scaling algorithm for Fenchel-type duality in discrete convex optimization. SIAM J. Optim. 13, 204–211 (2003)

    Article  MathSciNet  Google Scholar 

  • Iwata, S., Moriguchi, S., Murota, K.: A capacity scaling algorithm for M-convex submodular flow. Math. Program. 103, 181–202 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Jensen, P.M., Korte, B.: Complexity of matroid property algorithms. SIAM J. Comput. 11, 184–190 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Karzanov, A.V.: Concave cocirculations in a triangular grid. Linear Algebra Appl. 400, 67–89 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Klyachko, A.A.: Stable bundles, representation theory and Hermitian operators. Sel. Math. 4, 419–445 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Knutson, A., Tao, T.: Honeycombs and sums of Hermitian matrices. Not. Am. Math. Soc. 48, 175–186 (2001)

    MATH  MathSciNet  Google Scholar 

  • Knutson, A., Tao, T., Woodward, C.: The honeycomb model of GL n (C) tensor products II: Puzzles determine facets of the Littlewood–Richardson cone. J. Am. Math. Soc. 17, 19–48 (2003)

    Article  MathSciNet  Google Scholar 

  • Kobayashi, Y., Murota, K.: Induction of M-convex functions by linking systems. Discrete Appl. Math. 155, 1471–1480 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Kobayashi, Y., Takazawa, K.: Even factors, jump systems, and discrete convexity. METR 2007-36, Department of Mathematical Informatics, University of Tokyo (June 2007). J. Comb. Theory, Ser. B, to appear

    Google Scholar 

  • Kobayashi, Y., Murota, K., Tanaka, K.: Operations on M-convex functions on jump systems. SIAM J. Discrete Math. 21, 107–129 (2007)

    MATH  MathSciNet  Google Scholar 

  • Koichi, S.: The Buneman index via polyhedral split decomposition. METR 2006-57, Department of Mathematical Informatics, University of Tokyo (November 2006)

    Google Scholar 

  • Kolmogorov, V., Shioura, A.: New algorithms for the dual of the convex cost network flow problem with application to computer vision. Preprint (2007)

    Google Scholar 

  • Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Berlin (2008)

    Google Scholar 

  • Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55, 270–296 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Lovász, L.: Matroid matching and some applications. J. Comb. Theory, Ser. B 28, 208–236 (1980)

    Article  MATH  Google Scholar 

  • Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming—The State of the Art, pp. 235–257. Springer, Berlin (1983)

    Google Scholar 

  • Miller, B.L.: On minimizing nonseparable functions defined on the integers with an inventory application. SIAM J. Appl. Math. 21, 166–185 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  • Moriguchi, S., Murota, K., Shioura, A.: Scaling algorithms for M-convex function minimization. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 85-A, 922–929 (2002)

    Google Scholar 

  • Murota, K.: Valuated matroid intersection, I: optimality criteria. SIAM J. Discrete Math. 9, 545–561 (1996a)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K.: Valuated matroid intersection, II: algorithms. SIAM J. Discrete Math. 9, 562–576 (1996b)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K.: Convexity and Steinitz’s exchange property. Adv. Math. 124, 272–311 (1996c)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K.: Fenchel-type duality for matroid valuations. Math. Program. 82, 357–375 (1998a)

    MathSciNet  Google Scholar 

  • Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998b)

    MATH  MathSciNet  Google Scholar 

  • Murota, K.: Submodular flow problem with a nonseparable cost function. Combinatorica 19, 87–109 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000)

    MATH  Google Scholar 

  • Murota, K.: Discrete Convex Analysis—An Introduction. Kyoritsu Publishing Co., Tokyo (2001) (in Japanese)

    Google Scholar 

  • Murota, K.: Discrete Convex Analysis. SIAM Monographs on Discrete Mathematics and Applications, vol. 10. SIAM, Philadelphia (2003a)

    MATH  Google Scholar 

  • Murota, K.: On steepest descent algorithms for discrete convex functions. SIAM J. Optim. 14, 699–707 (2003b)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K.: Note on multimodularity and L-convexity. Math. Oper. Res. 30, 658–661 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K.: M-convex functions on jump systems: A general framework for minsquare graph factor problem. SIAM J. Discrete Math. 20, 213–226 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K., Shioura, A.: Extension of M-convexity and L-convexity to polyhedral convex functions. Adv. Appl. Math. 25, 352–427 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K., Shioura, A.: Conjugacy relationship between M-convex and L-convex functions in continuous variables. Math. Program. 101, 415–433 (2004a)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K., Shioura, A.: Fundamental properties of M-convex and L-convex functions in continuous variables. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 87-A, 1042–1052 (2004b)

    Google Scholar 

  • Murota, K., Tamura, A.: On circuit valuation of matroids. Adv. Appl. Math. 26, 192–225 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Murota, K., Tamura, A.: Proximity theorems of discrete convex functions. Math. Program. 99, 539–562 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Narayanan, H.: Submodular Functions and Electrical Networks. Annals of Discrete Mathematics, vol. 54. North-Holland, Amsterdam (1997)

    MATH  Google Scholar 

  • Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions I. Math. Program. 14, 265–294 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992)

    MATH  Google Scholar 

  • Recski, A.: Matroid Theory and Its Applications in Electric Network Theory and in Statics. Springer, Berlin (1989)

    Google Scholar 

  • Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  • Roth, A.E., Sotomayor, M.A.O.: Two-Sided Matching—A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  • Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  • Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  • Shapley, L.S., Shubik, M.: The assignment game I: The core. Int. J. Game Theory 1, 111–130 (1972)

    Article  MathSciNet  Google Scholar 

  • Shioura, A.: Fast scaling algorithms for M-convex function minimization with application to the resource allocation problem. Discrete Appl. Math. 134, 303–316 (2003)

    Article  MathSciNet  Google Scholar 

  • Shioura, A.: On the pipage rounding algorithm for submodular function maximization: A view from discrete convex analysis. METR 2008-03, Department of Mathematical Informatics, University of Tokyo (January 2008)

    Google Scholar 

  • Shioura, A., Tanaka, K.: Polynomial-time algorithms for linear and convex optimization on jump systems. SIAM J. Discrete Math. 21, 504–522 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Sotomayor, M.: A labor market with heterogeneous firms and workers. International J. Game Theory 31, 269–283 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Speyer, D.: Tropical linear spaces. arXiv:math.CO/0410455 (2004)

  • Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4, 389–411 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32, 41–43 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Tamura, A.: Applications of discrete convex analysis to mathematical economics. Publ. Res. Inst. Math. Sci. 40, 1015–1037 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Tamura, A.: Coordinatewise domain scaling algorithm for M-convex function minimization. Math. Program. 102, 339–354 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Topkis, D.M.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998)

    Google Scholar 

  • van der Laan, G., Talman, D., Yang, Z.: Solving discrete zero point problems. Math. Program. 108, 127–134 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: 40th ACM Symposium on Theory of Computing (May 2008)

    Google Scholar 

  • White, N. (ed.): Theory of Matroids. Cambridge University Press, London (1986)

    MATH  Google Scholar 

  • Zipkin, P.: On the structure of lost-sales inventory models. Oper. Res. (2008). doi:10.1287/opre.1070.0482

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuo Murota

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  1. Kazuo Murota
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Editors and Affiliations

  1. Industrial and Systems Engineering, Georgia Tech, 765 Ferst Drive, Atlanta, Georgia, 30332-0205, USA

    William Cook

  2. Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

    László Lovász

  3. Research Institute for Discrete Mathematics, University of Bonn, Lennéstr. 2, 53113, Bonn, Germany

    Jens Vygen

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Murota, K. (2009). Recent Developments in Discrete Convex Analysis. In: Cook, W., Lovász, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_11

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